REGULARITY PROPERTIES OF FUNCTIONAL EQUATIONS IN … · 2013-07-18 · PREFACE This book is about...

REGULARITY PROPERTIES OF FUNCTIONAL EQUATIONS IN SEVERAL VARIABLES

Advances in Mathematics

VOLUME 8

Series Editor:

J. Szep, Budapest University of Economics, Hungary

Advisory Board:

S.-N. Chow, Georgia Institute of Technology, U.S.A.

G. Erjaee, Shiraz University, Iran

W. Fouche, University of South Africa, South Africa

P. Grillet, Tulane University, U.S.A.

H.J. Hoehnke, Institute of Pure Mathematics of the Academy of Sciences, Germany

F. Szidarovszky, University of Airzona, U.S.A.

P.G. Trotter, University of Tasmania, Australia

P. Zecca, Universitb di Firenze, Italy

REGULARITY PROPERTIES OF FUNCTIONAL EQUATIONS IN SEVERAL VARIABLES

ANTAL JARAI Eotvos Lorand University, Budapest, Hungary

Springer -

Library of Congress Cataloging-in-Publication Data

A C.I.P. record for this book is available from the Library of Congress.

ISBN 0-387-24413-1 e-ISBN 0-387-24414-X Printed on acid-free paper.

O 2005 Springer Science+Business Media, Inc.

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden.

The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.

Printed in the United States of America.

9 8 7 6 5 4 3 2 1 SPIN 11 377986

This book is dedicated to JBnos Aczkl, my "mathematical grandfather", the teacher of several of us in the field of functional equations, and to my teacher ZoltAn Dar6czy who introduced me to functional equations.

TABLE OF CONTENTS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . Preface ix

. . . . . . . . . . . . . . . . . . Chapter I . PRELIMINARIES 1 . . . . . . . . . . . . . . . . . . . . . . . . $1 . Introduction 1

. . . . . . . . . . . . . . . . . . $ 2 . Notation and terminology 40

. . . . . . . . . . . . Chapter I1 . STEINHAUS TYPE THEOREMS 53

. . . . . . . . . . . $ 3 . Generalizations of a theorem of Steinhaus 53 . . . . . . . . . . . . 5 4 . Generalizations of a theorem of Piccard 66

. . . . Chapter I11 . BOUNDEDNESS AND CONTINUITY OF SOLUTIONS 73 . . . . . . . . . . . . . . . . $ 5 . Measurability and boundedness 73

. . . . . . . . . . $ 6 . Continuity of bounded measurable solutions 76 . . . . . . . . . . . . . . . . . . . 5 7 . On a problem of Mazur 81

. . . . . . . . . . . . . . 5 8 . Continuity of measurable solutions 86 . . . . . . . . . $ 9 . Continuity of solutions having Baire property 94

. . . . . . . . . . . . . . . . . . . . . $10 . Almost solutions 100

. . . . . . . Chapter IV . DIFFERENTIABILITY AND ANALYTICITY 109

. . . . . . . 5 11 . Local Lipschitz property of continuous solutions 109 . . . . . . . . . . . . . . . 5 12 . Holder continuity of solutions 128 . . . . . . . . . . . . . . . $ 13 . Solutions of bounded variation 132

. . . . . . . . . . . . . . . . . . . . . $ 14 . Differentiability 137 . . . . . . . . . . . . . . . . 5 15 . Higher order differentiability 141

. . . . . . . . . . . . . . . . . . . . . . . 5 16 . Analyticity 144

. . . . . . Chapter V . REGULARITY THEOREMS ON MANIFOLDS 157

. . . . . . . . . . . . 5 17 . Local and global results on manifolds 157

. . . Chapter VI . REGULARITY RESULTS WITH FEWER VARIABLES 169 . . . . . . . . . . . . . . . . . . . . 5 18 . ~wiatak 's method 170

. . . . . . . . . . . . § 19 . Between measurability and continuity 174 . . . . . . . . . . . 5 20 . Between Baire property and continuity 204 . . . . . . . . . . . $ 21 . Between continuity and differentiability 218

. . . . . . . . . . . . . . . . Chapter VII . APPLICATIONS 231 . . . . . . . . . . . . . . . . . . . . 5 22 . Simple applications 231

. . . . . . . . . $23 . Characterization of the Dirichlet distribution 275 . . . . . . . $24 . Characterization of Weierstrass's sigma function 285

viii Table of contents

References . . . . . . . . . . . . . . . . . . . . . . . . . . 335

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

PREFACE

This book is about regularity properties of functional equations. It contains, in a unified fashion, most of the modern results about regularity of non-composite functional equations with several variables. It also contains several applications including very recent ones. I hope that this book makes these results more accessible and easier to use for everyone working with functional equations.

This book could not have been written without the stimulating atmo- sphere of the International Symposium on Functional Equations conference series and thus I am grateful to all colleagues working in this field. This series of conferences was created by JBnos Aczitl. I am especially grateful to him for inviting me to the University of Waterloo, Canada, which provided a peaceful working environment. I started this book in September 1998 during my stay in Waterloo.

I thank Mikl6s Laczkovich, the referee of my C.Sc. dissertation very much for his remarks and suggestions.

Between 1974 and 1997 I worked at Kossuth Lajos University, Debrecen, Hungary. Naturally, I am grateful to all members of the Debrecen School of functional equations.

Finally, I would like to thank those of my colleagues, joint pieces of work with whom in one way or another are contained in this book: JAnos Aczkl, ZoltBn Darbczy, Roman Ger, Gyula Maksa, Zsolt PAles, Wolfgang Sander, and LAsz16 Szitkelyhidi.

I thank my sons, Antal JArai Jr. and ZoltAn JBrai for correcting several errors of my "Hunglish"; all the remaining errors are mine.

Finally, I wish to express my gratitude to Kluwer Academic Publishers, in particular, Professor Jen6 Sz6p and John C. Martindale for the first rate and patient technical help.

Financial support for this book was provided mainly by Szitchenyi Schol- arship for Professors, Hungary and partly by OTKA TO31995 grant.

Budapest, May 25, 2003. Antal JBrai

Department of Computer Algebra Eotvos LorAnd University

PBzmAny Pitter sittAny 1/C H-1117 Budapest, Hungary

e-mail: [email protected] http://compalg.inf.elte. hu/-ajarai

Chapter I.

PRELIMINARIES

In this chapter, starting with simple examples, we describe the problems with which we will deal in this book. We also present simple examples of our methods. First we formulate the fundamental problem, then analyse its conditions and explore its applicability. We then formulate theorems that follow from our results as corollaries to that fundamental problem. Then we survey possibilities for generalization. We close this chapter by summarizing our notation and terminology, including the formulation of theorems not readily available in the literature or usually formulated in a different way.

1. INTRODUCTION

1.1. General considerations a n d simple examples. As a first, illustrative example let us consider the best-known functional equation, Cauchy's equation

with unknown function f . In a wider sense differential equations, integral equations, variational problems, etc. are also functional equations, but here we will use this expression in a more restrictive sense for functional equations

2 Chapter I. Preliminaries

without infinitesimal operations such as integration and differentiation. For a more formal definition, see Aczd [3], 0.1. To formulate a functional equation exactly we have to give the set of functions in which we look for solutions. We also have to give the domain of the functional equation. In the above example this is the set of the pairs (x, y) of the variables x and y for which equality has to be satisfied. For example, we may look for all measurable functions f : R -+ R such that (1) is satisfied for all (x, y) E [0, m[ x R. Conditions such as measurability, Baire property, continuity everywhere or in a point, boundedness, differentiability, analyticity, etc. are called regularity conditions. If this kind of conditions are imposed on the solution, then we say that we look for regular solutions. Otherwise, if we look for solutions among all maps from a given set into another given set, then we say that we look for the general solution of the functional equation.

Usually, the domain of the functional equation is the set of all tuples of the variables for which both sides are defined. For example, if we say that f : R -+ R is a solution of Cauchy's functional equation, then it is implicitly understood that (1) is satisfied for all (x, y) E R x R. If the domain of the equation is not the largest possible for which both sides are defined, then we speak about an equation with restricted domain; the term conditional equation is also used, especially if the domain of the equation also depends on the solution or solutions.

Cauchy's equation is a functional equation with two variables; the variables denoted by x and y in (1). Equations like f (x) = f (-x), f (x) = - f (-x), f (22) = f ( x ) ~ , or difference equations are called functional equations in a single variable. The "single variable" may also be a vector variable; it is understood that there are no more variables in the equation than the number of places in the unknown function or the minimal number of places in the unknown functions - if there is more than one. Otherwise we speak about a functional equation in several variables. This distinction is very useful in practice. There is a large difference between functional equations with a single variable and several variables: the methods used in the two cases are quite different. In this book we deal with functional equations in several variables. About equations in a single variable see the books Kuczma [I261 and Kuczma, Choczewski, Ger [128].

The distinction between functional equations in a single variable and in several variables and what we have said about variables, domain, regular and general solutions also apply to systems of functional equations.

Further simple examples of functional equations are Cauchy's exponen-

§ 1. Introduction 3

tial equation

Cauchy's power equation

and Cauchy's logarithmic equation

Observe, that solutions f of (2) mapping 10, GO[ into the normed algebra of all bounded linear operators on a Banach space gives operator semigroups. The usefulness of semigroups in the study of evolution equations such as the heat equation or Schrodinger's equation is well known, see for example Hille and Phillips [72]. The overall importance of equations (1)-(4) is due to the fact that they describe homomorphisms.

We move toward a general theory of functional equations, and we do not intend to study specific functional equations, except as examples, even if they are very important.

It is a well-known phenomenon that one functional equation can determine several unknown functions. This is the situation, for example, for the analogue of Cauchy's functional equation with several unknown functions which is called Pexider's equation:

Indeed, if f 1, f2 , f3 : R -+ R, by putting y = 0 and x = 0 in (5) we may express f2 and f3 by f l , respectively. By putting x = 0 and y = 0 simultaneously in (5) and using the resulting relation we obtain that f = f l - f1 (0) satisfies Cauchy's equation (1). Hence (5) can be reduced to (1). Similar phenomena occur often when different occurrences of the unknown function f are replaced by f l , f2 , etc., a process sometimes called "Pexiderization".

Jensen 's equation

can be considered as a special case of Pexider's equation, and we obtain that an f : R -+ R function is a solution of (6) if and only if the function f - f (0) satisfies (1).


It is also possible to consider functional inequalities. Functional inequality

related to Cauchy's equation describes subadditive functions and functions satisfying

are the so-called Jensen convex functions. We will use the above simple functional equations as illustrative exam-

ples. Their detailed study can be found in the book of Acz6l [3] or in the book of Acz6l and Dhombres [20].

1.2. Simple examples: smooth solutions. Let us suppose that a solution f : IR + R of Cauchy's equation f (x + y) = f (x) + f (y) is analytic. Substituting y = x we obtain the equation f (22) = 2 f (x) in a single variable x E R. Analyticity is such a strong regularity condition that even this single variable equation has not too many analytic solutions. For the solution f (x) = co + clx + we obtain

in a neighborhood of the origin, and hence that the solution can only be f (x) = cx with an arbitrary constant c = cl. Substitution shows that this is indeed a solution of Cauchy's equation.

The case of Cauchy's exponential equation f (x + y) = f (x) f (y) is much more interesting. As above, we obtain the single variable equation f (22) = f (x)~, and, if f : R + R is analytic, f (x) = co + clx + 3 . . , then

Hence co = ci. There are two possibilities. The first is that co = 0, which implies that c, = 0 for each n, and hence f = 0. The second is that co = 1. In this case cl could be arbitrary, and from the equation

$1. Introduction 5

using the notation c = cl, we obtain by induction that cn = cn/n!. Hence all analytic solutions are given by f (x) = C r = o cnxn/n! = exp(cx). The same method gives complex analytic solutions f : C + C, too. Let us observe that this is a nice way to introduce exponential functions (and hence the related functions sin, cos, sinh, and cosh) using only the most important property of exponentiation. Note that this was Cauchy's original motivation to investigate functional equations 1.1. (1)-1.1. (4) : he wanted to avoid "circulus vitiosus" by studying power functions; see the historical remarks in the book of Acz6l and Dhombres [20], pp. 365-371.

Now let us only suppose that the solution f : R + R is twice differentiable. In the case of Cauchy's equation, f (x + y) = f (x) + f (y), let us differentiate both sides with respect to y. This "kills" the first term on the right-hand side, and we obtain that f l ( x + y) = f '(y) for every x, y E R. Differentiating again, but with respect to x we can "kill" the other term on the right-hand side, too, and we obtain f"(x + y) = 0. Substituting y = 0 we have f "(x) = 0, a differential equation. All solutions of this equation have the form f (x) = co + cx, co, c E R. Substituting this into the original functional equation we see that co = 0, and we obtain that twice differentiable solutions are exactly the functions f (x) = cx.

This simple example illustrates a general method to get "smooth enough" solutions. The general tactic is to "kill" some terms by applying appropriate differential operators, and to obtain differential equations by appropriate substitutions. Usually, appropriate substitutions or use of certain symmetries of the equation results in a differential equation with lower degree. For example, substituting y = 0 in the equation f l ( x + y) = fl(y) we obtain that f l (x ) = c with c = fl(0), a first order equation. Cauchy's exponential equation, f (x + y) = f (x) f (y), similarly yields f '(x + y) = f (x) f '(y), and after substituting y = 0 we obtain f '(x) = cf (x), where c = f '(0).

Let us observe that in both cases, the general once differentiable solution f : R + R is the same as the general analytic solution.

1.3. Simple examples: regularity properties. How to obtain solutions of the above examples, Cauchy's equation and Cauchy's exponential equation under much weaker regularity assumptions? A general way is to prove that weak regularity conditions, say continuity or measurability of solutions implies much stronger regularity conditions, their differentiability or even analyticity. For example, let us observe that in both of the above cases the differential equation obtained for the solutions in the previous point implies directly that the solutions are analytic (see Dieudonnh [49], 10.5.3).

If we have a continuous solution f : R + R, then integrating Cauchy's


equation over an interval [a, b] of positive length yields

Substituting a new variable u = x + y we obtain

The right-hand side is differentiable, so we obtain that f is differentiable. If we want to deduce that f is twice differentiable, we can apply the same reasoning to the equation f '(x + y) = f '(x) obtained by differentiation with respect to x from the original. Higher order differentiability can be obtained analogously.

In the case of Cauchy's exponential equation f - 0 is one of the continuous solutions f : R + R. If f (yo) # 0, then we can choose a neighborhood [a, b] of yo such that f (y)/ f (yo) 2 112 for each y E [a, b]. Integrating we obtain

and hence that

This implies that f is differentiable. Here, again, applying the same method for the equation fl(x + y) = fl(x) f (y) obtained from the original equation by differentiation with respect to x gives that the solutions are twice differentiable, etc.

Now, let us consider a measurable solution f : R -+ R of Cauchy's equation. Let [a, b] be an interval with positive length 7. Let xo E R be arbitrary. By Lusin's theorem there exists a compact set C1 contained in [xo + a, xo + b] and having Lebesgue measure greater than 3714 such that f lC1 is continuous. If /x - xo/ < 7718, then the set C1 - x is contained in C = [a - 718, b + 7/81. Since the Lebesgue measure of C \ (C1 - x) and C \ (C1 - xo) are less than 712, they cannot cover C. Hence the intersection (C1 - x) n (C1 - xo) is nonvoid. Now, let E > 0 be arbitrary. Since f lCl is uniformly continuous, there exists a S > 0 such that if u, u' E C1 then If (u) - f (.')I < E. Hence, if Ix - xol < min{71/8, S) then for any y E (C1 - x) n (CI - xo) we obtain

'$1. Introduction 7

i. e., f is continuous at xo. Since x0 was arbitrary, f is continuous everywhere. The same method can be applied to Cauchy's exponential equation after introducing the new variable t = x + y instead of x, i. e., to the equation

f (4 = f (t - df (Y). Note that Cauchy's logarithmic equation f (xy) = f (x) + f (y) has no

other solution f : R + R than f = 0; this follows by substituting y = 0. In the case of Cauchy's power equation f (xy) = f (x) f (y) there are

solutions f : R + R which are measurable but non-continuous, continuous but non-differentiable, etc. Indeed, the functions x e /x 1 and x ci\ 1x1 sgn x are solutions for any c E R if OC is understood as 0.

1.4. Hilbert 's fifth problem. In his celebrated address to the 1900 International Congress of Mathematicians, in his fifth problem Hilbert ([70] p. 304) asked1

". . . how far Lie's concept of continuous groups of transformations is approachable in our investigations without the assumption of differentiability of the functions"

More precisely2:

". . . hence there arises the question whether, through the introduction of suitable new variables and parameters, the group can always be transformed into one whose defining functions are differentiable

>, . . . .

Explaining that the group property is connected to a system of functional equations, in the second part of his fifth problem Hilbert goes on as follows3:

". . . inwieweit der Liesche Begriff der kontinuierlichen Transformationsgruppe auch ohne Annahme der Differenzierbarkeit der Funktionen unserer Untersuchung zuganglich s t . "

". . . es ensteht mithin die Rage, ob nicht etwa durch Einfuhrung geeigneter neuer Verandertlicher und Parameter die Gruppe stets in eine solche iibergefiihrt werden kann, fiir welche die definierenden Funktionen differenzierbar sind, . . . " .

" ~ b e r h a u ~ t werden wir auf das weite und nicht uninteressante Feld der Funktional- gleichungen gefuhrt, die bisher meist nur unter der Voraussetzung der Differenzierbarkeit der auftretenden Funktionen untersucht worden sind. Insbesondere die von ABEL (Werke, Bd. 1, S. 1, 61, 389) mit so vielem Scharfsinn behandelten Funktionalgleichungen, die Dif- ferenzengleichungen und andere in der Literatur vorkommende Gleichungen weisen an sich nichts auf, was zur Forderung der Differenzierbarkeit der auftretenden Funktionen zwingt, und bei gewissen Existenzbeweisen in der Variationsrechnung fie1 mir direkt die Aufgabe zu, aus dem Bestehen einer Differenzengleichung die Differenzierbarkeit der betrachteten Funktionen beweisen zu mussen. In allen diesen Fallen erhebt sich daher die Frage, inwieweit etwa die Aussagen, die wir im Falle der Annahme differenzierbarer Funktionen machen konnen, unter geeigneten Modifikationen ohne diese Voraussetzung giiltig sind."


"Moreover, we are thus led to the wide and interesting field of functional equations which have been heretofore investigated usually only under the assumption of the differentiability of the functions involved. In particular the functional equations treated by Abel (Oeuvres, vol. 1, pp. 1, 61, 389) with so much ingenuity, the difference equations, and other equations occurring in the literature of mathematics, do not directly involve anything which necessitates the requirements of the differentiability of the accompanying functions. In the search for certain existence proofs in the calculus of variations I came directly upon the problem: To prove the differentiability of the function under consideration from the existence of a difference equation. In all these cases, then, the problem arises: In how far are the assertions which we can make in the case of differentiable functions true under proper modifications without this assumption ?"

(Hilbert's emphases.) After this Hilbert quotes a result of Minkowski which states that under certain conditions the solutions of the functional inequality

are partially differentiable, and remarks that certain functional equations, for example the system of functional equations

where a, p are given real numbers, may have solutions f which are continuous but non-differentiable, even if the given function g is analytic.

In our present-day language, it is customary to formulate the fifth problem of Hilbert as the question whether a locally Euclidean topological group is a Lie group. However, in the second part of his fifth problem, Hilbert draws attention to more general problems which today are called regularity problems. They require to prove that differentiability assumptions for functional equations, differential equations, and other equations can be replaced by much weaker assumptions (possibly with appropriate modifications of the problem). This idea returns in problems nineteen and twenty of Hilbert concerning calculus of variation and partial differential equations. See the book of Zeidler [209], II/A, pp. 86-93. As a general reference about Hilbert's problems, see the book [26] edited by Alexandrov.


Concerning Abel's work on functional equations mentioned by Hilbert see the short note [12] and the paper [14] of Aczkl. These initiated intense research to solve Abel's equations under as weak conditions as possible. Two of these equations will be used here as illustration to our methods among applications (see 22.10 and 24); concerning a third one see the papers of Sablik 11 741, 11 751, [176].

1.5. A general scheme t o solve functional equations. The simple examples above, as well as the second part of Hilbert's fifth problem, suggest a general scheme to obtain regular solutions of functional equations:

(1) Prove - using regularity theorems - that each solution of the functional equation satisfying some "weak" regularity property - say measurability, the Baire property, etc. - satisfies "strong" regularity conditions - for example, local integrability, continuity, differentiability or analyticity.

(2) Obtain the "strongly" regular solutions by deriving another kind of equation from the functional equation (differential equation, integral equation, an equation system for the coefficients of the power series expan- sion, etc.) or by using special methods of the theory of functional equations.

Hundreds of papers illustrate this time-honored practice. In step (2) we use special properties of the functional equations and ad hoc considerations: special substitutions, symmetries of the equations, integral transforms, "killing" some terms by applying appropriate differential operators, etc. This is practiced in several books and papers about functional equations. The reader can find some examples among the applications in the last chapter of this book. It is impossible to refer here to all the enormous literature. We refer the reader to books, survey papers, and research papers. The following books contain introductory material about functional equations in several variables: Aczd [2], [5]; Hille [71]; Kuczma [127]; Saaty [173], section 3. The following are monographs about functional equations in several variables: Acz4l [I], and the revised English edition [3]; Aczd-Dhombres [20]. Special topics are dealt with in Acz6l [lo] (social and behavioral sciences); Aczkl-Dar6czy 1191 (measures of information); Aczkl-Golqb [22] (geometry); Dhombres [45] (conditional Cauchy equations); Eichhorn [51] (economics); Szekelyhidi [I931 (convolution-type functional equations).

See moreover the survey papers [8], [14], [16], [120], the survey papers in the volume [15], the volumes of Aequationes Mathematicae, and the bibliography of papers on functional equations in the volumes of Aequationes Mathematicae.


The main subject of this book is to deal with step (1). We will obtain general regularity results for a large class of functional equations, called non-composite functional equations with several variables. Our results give answers to most of the questions of the second part of Hilbert's fifth problem for this class. Here it is also impossible to refer to all the papers dealing with the regularity problem of special functional equations. We refer the reader to the book and survey papers cited above and to the volumes of Aequationes Mathematicae and the bibliography therein. Part of the regularity results dealing with general types of functional equations in several variables will be cited in the introduction of the appropriate sections of this book.

Before starting to deal with regularity questions, we give a short overview of and some references about general methods to get "smooth" solutions of functional equations.

1.6. General methods of determining "smooth" solutions of functional equations. In 1.2 we have seen two methods. One of them was to find analytic solutions by obtaining a system of equations for the coefficients from the functional equation. The other one was to reduce the functional equation to a differential equation. This second method, which is used extensively, usually gives the most comfortable way to determine "smooth" solutions. Examples can be found in the book of Aczd [3], see especially 4.2.1, 4.2.4. The deduction of the differential equation usually uses the tactic of "killing some terms" by differentiation, and using special substitutions and special symmetries of the equation. Actually this method goes back to Abel. He wrote (quote from Aczkl [9]):4

"Actually, as many equations can be found by repeated differentia- tions with respect to the two independent variables as are necessary to eliminate arbitrary functions. In this manner, an equation is obtained which contains only one of these functions and which will generally be a differential equation of some order. Thus it is generally possible to find all the functions by means of a single equation.

"In der Tath lassen sich durch wiederholte Differentiationen nach den beiden veran- dertlichen Grossen, so vie1 Gleichungen finden, als notig sind, um belibige Functionen zu eliminieren, so dass man zu einer Gleichung gelangt, welche nur noch eine dieser Functionen enthalt und welche im Allgemeinen eine Differential-Gleichung von irgend einer Ordnung sein wird. Man kann also im Allgemeinen alle die Functionen vermittelst einer einzigen Gleichung finden. Daraus folgt, dass eine solche Gleihung nur selten moglich sein wird. Denn da die Form einer beliebigen Function die in der gegebenen Bedingungs-Gleichung vorkommt, vermoge der Gleichung selbst, von der Formen der andern abhangig sein soll, so ist offenbar, dass man im Allgemeinen keine dieser Functionen als gegeben annehmen kann."

5 1. Introduction 11

From this it follows that such an equation can exist only rarely. Indeed, since the form of an arbitrary function appearing in the given conditional equation, by virtue of the equation itself, is to be dependent on the forms of others, it is obvious that, in general, one cannot assume any of these functions to be given."

For functional equations with several variables and several unknown functions PAles [163] developed this method into a general algorithm. We will treat his algorithm in detail in 5 16 and we will illustrate it in 22.11. Presently this is the most general method to find smooth solutions.

Another important method is to use an integral transform to get a simpler functional equation. This method is illustrated in the book of Aczkl [3], 4.1.1. Szkkelyhidi developed this idea into a very general method. His books [191], [I941 contain several examples, mostly functional equations on commutative groups, where mean periodic functions, Fourier transform and some refinements and spectral synthesis play an important rde; see also our joint survey paper [120]. With Szkkelyhidi's methods various special cases of the general functional equation

can be treated, where functions f j , gi, hi are unknown and functions cpj , y ! ~ ~ are given. In general, cp j , $j are homomorphisms, or in some sense structure- preserving mappings of the common domain of the unknown functions. G is a commutative topological group or semigroup. For simplicity we restrict ourselves here to the case where the range of the unknown functions is in C, although more general ranges can be considered as well. Equation (1) can be roughly characterized as a "linear equation with linear arguments", although it is linear only in the sense that linear combinations of solution vectors (f l , . . . , f,) also satisfy a similar equation with another n. Although the methods and results depend heavily on the commutative structure of the underlying groups or semigroups, in some cases they can be applied also in the noncommutative situation.

An important special case is n = 0, where regular solutions are continuous (generalized) polynomials, i. e., they can be represented as a finite sum x ++ Ck A,+(x, x, . . . , x) where A,+ : G" C is a k-additive function, i. e., additive in each variable. The exact characterization of the solutions depends on some algebraic conditions.

Other important special cases are the Levi-CivitA type equation (case m = 1, cpl = $1 = id) and the D'Alembert-type equation (case m = 2,


~1 = $1 = ( ~ 2 = -g2 = id). In these cases regular solutions f l and f l , f2, respectively, are continuous exponential polynomials, i. e., can be represented as a finite sum pkmk, where each p,+ : G -+ C is a (generalized) polynomial and each mk : G -+ C is a multiplicative function.

All these special cases are related to systems of convolution-type functional equations f *,u = 0, where p runs in a given set of compactly supported complex measures. The essence of the method for solving such convolution- type systems is to prove that the continuous solutions form a translation invariant closed linear subspace, and there are "sufficiently many" exponential polynomials in this subspace. The continuous solutions can be constructed from exponential polynomial solutions.

All these methods work for general solutions, too, if we consider G with the discrete topology.

The elementary method of SGto is based on the following observation:

Let X, Y be sets, F; : X + IF, gi : Y + IF (i = 1,2 , . . . , n) be mappings into a field. Suppose, that the functional equation

n

(2) 0 = x fi (x)gi (y) for all x E X , y E Y i=l

is satisfied. I f f l , f2 , , . . . , f, are linearly independent and f j = El='=, ci,j f i for j = r + 1,. . . , n with some ci,j E IF, then gi = - Cy=r+l ci,jgj for i = 1 , 2 , . . . , r . Especially,

Indeed,

and since f l , . . . , f, are linearly independent, their coefficients have to be zero for any y E Y.

We remark that the above statement is a generalization of the frequently used trivial observations that if f (x)g(y) 0, then f (a) G 0 or g(y) - 0 and that if f (x) = g(y) for all x E X, y E Y, then there exists a constant c such


that f (x) - c and g(y) = c. These correspond to the case n = 1 and n = 2, respectively.

Of course, the above statement is a special case of more general theorems. We mention it here explicitly only because several nonlinear or even composite functional equations can be solved by Suto's method, reducing them - by differentiation or by other tricks - to equation (2). See for some recent applications of S ~ t o ' s method the papers [144-1501 of Lundberg. We will give a simple example of how to apply Suto's method in 22.11.

Certain functional equations can be reduced to integral equations. We do not illustrate this method here; see the book of Aczd [3], 4.1.2.

Another method, which needs only weak regularity conditions, the so- called "distribution method" was initiated by Fenyo [54]. The main idea is that continuous (or locally integrable) solutions are considered as Schwartz distributions and the equation is viewed as an equation for distributions. Differential equations (in the distribution sense) are obtained using similar tricks as for smooth solutions, but usually with more technical problems. Fi- nally, distribution solutions turn out to be regular distributions belonging to (smooth) functions. See the papers of Baker [31], [32], [33], [34] about this method. The applicability of the distribution method is restricted, because no multiplication among Schwartz distributions is defined. By Schwartz's impossibility theorem, this cannot be done in a satisfying way. It is even more hopeless to substitute distributions into general Cm functions with several variables. The distribution method has to be restricted to functional equations that are not very far from being linear.

Sometimes it is possible to find the solution of a functional equation directly, on a dense set. In such cases continuous solutions are determined by solutions on this dense set. For example, for Cauchy's equation, by substitution x = y = 0 we obtain f (0) = 0 and by induction f (nx) = n f (x) for n E N. Substitution y = -x shows that f (-x) = -f (x), hence the above formula holds for all n E Z. Now substituting x l n for x shows that f (xln) = f (%)In, whenever 0 # n E Z. Hence f (rx) = r f (x), whenever r E Q. This shows that f (1) determines the values of f at rational points and by approximating real numbers with rational numbers we obtain that the only continuous solutions are f (x) = cx where c = f (1). For further examples see the book of Aczdl [3], section 2.

A further method is to reduce the functional equation to another functional equation which is already solved. We have seen an example for this in 1.1. In this more or less algebraic way we can usually find the general


solutions of the equation. Usually, regular solutions of the original equation are obtained from regular solutions of the other equation, hence we may determine the regular solutions, too. The reader can find several examples for this kind of reduction of a functional equation to another functional equation in Aczkl [3] and in Aczkl-Dhombres [20]. We note that merely the knowledge of the general solution does not always make it possible to deduce the regular solutions easily. For example, the general solution of Cauchy's equation is the following: let us fix an arbitrary base B of R over Q, a Hamel base. Then every real number x can be uniquely written as finite linear combinations xi ribi of base elements hi with rational coefficients ri. Any function f : B -+ R can be uniquely extended to a solution of Cauchy's equation by f (x) = Ci ri f ( h i ) Let us observe that even in this very simple case it is not clear - without regularity results - which solutions are measurable.

We have to mention here the method of Vincze. It is based on the following observation:

Suppose that functions fi l j , i = 1,2, . . . , m, j = 1,2, . . . , n map an arbitrary set X into a field IF and they satisfy the functional equation

Then for any function fo : X + IF the functional equation

is satisfied, too, where fi,0 = fo for all i = 1,2, . . . , m .

The statement can be easily verified by expanding all determinants in (4) with respect to fO(xk) , k = 0, I , . . . , n.

By choosing an appropriate fo some terms may become zero and we get a simpler functional equation. For further information on how to apply this method and how to reduce certain functional equations to the form (3) see the paper of Vincze [205].

We conclude this section with some general remarks. For general purposes usually the simplest way is to use general regularity results to prove that the solutions are a few times differentiable, and then to obtain a differential equation for the solutions. We usually cannot completely avoid using regularity results with the other methods either. Methods that use


less regularity are usually more complicated and cannot be applied to the most general cases. Despite general methods, we are very far from having algorithmic methods to solve a large part of the functional equations that are of recent interest.

1.7. Composite and non-composite functional equations with several variables. The main topic of the present book is to prove general regularity results. For several functional equations regularity theorems do not hold. Besides functional equations in a single variable - like those mentioned by Hilbert, see 1.4 - there are also several composite functional equations (in which unknown functions are substituted into each other) that have continuous, but non-differentiable solutions. For example, the Acz& Benz functional equation

first studied in [17], has the solution x H i(n: * 1x1). All continuous solutions of this equation were given by Dar6czy [41]. The functional equation system describing group axioms in the first part of Hilbertls fifth problem also needs special treatment (re-coordinating). Although, for example, in the paper of Acz6l [4] under very general conditions it is proved that the real function solutions f of the functional equation

are monotonic, it seems to us that there is no general regularity theory yet for such equations. (We will summarize some recent results about composite equations in 1.16.) However, for those functional equations with several variables where the unknown functions are not substituted into themselves or into each other, general regularity theorems can be proven. Such theorems constitute the main body of this book.

Our purpose is to investigate the "most general" non-composite functional equation with several variables having the form

for all (X, Y) from a set E. Here f , f o , f l l . . . , fn are unknown functions, all other functions are given. All functions are vector-vector functions. In- troducing the new variable x = G(X, Y) and solving the above equation for


f (x) we usually get a somewhat simpler form, which much better suits our purposes. In several cases introducing another new variable y = Go(X, Y), we can get the form

where in one of the unknown functions only the variable y appears. Since for our purposes this "explicit" equation is the most convenient one, we will use this form in general and we do not deal with the conditions under which the above transformations can be (at least locally) performed.

In this equation it seems to be unnecessary to include fa - and hence also to introduce the new variable y = Go(X, Y) -, but fo has a distin- guished r61e among the unknown functions on the right-hand side. As a problem of Sander [I821 shows (see 22.2), it is often enough to suppose much weaker conditions (or none at all) for f than for the other unknown functions. For the given functions we mainly have to prescribe smoothness conditions: simple examples show that these conditions cannot be omitted. Of course, we may suppose, that the right-hand side of the equation really depends on y, and so the equation is not an equation in a single variable. This is certainly

agi the case if we suppose that the rank of the matrices - is maximal. We

usually suppose that the set D is an open set. 89

A general problem of regularity is to investigate if Lebesgue measurable solutions or solutions having the Baire property are analytic. Most of our regularity theorems state that, if the given functions are "sufficiently nice" and the unknown functions f l , fP , . . . , fn have a regularity property, then the function f has a regularity property which is "one better".

To obtain an affirmative answer to the regularity problem, the following steps can be used:

(I) measurability implies continuity;

(11) solutions having the Baire property are continuous;

(111) continuous solutions are almost everywhere differentiable;

(IV) almost everywhere differentiable solutions are continuously differentiable;

(V) all p times continuously differentiable solutions are p + 1 times continuously differentiable;

(VI) infinitely many times differentiable solutions are analytic.


If f = fo = f l = . . . = f n r then we obtain step by step that the measurable solutions, or solutions having the Baire property, are analytic. If this is not the case, then expressing other unknown functions from the equation might help to draw a similar conclusion.

Let us remark that the regularity problem of composite and non-composite functional equations overlap. Indeed, calling functions "unknown" and "known" in regularity theorems is done only to distinguish those functions for which very weak regularity assumptions (say measurability) are supposed from those for which much stronger assumptions (say continuous differentiability) are supposed. Hence weakening regularity assumptions on "known" functions yields results much more useful for composite equations.

To better understand why the above steps (I)-(VI) are chosen, what the main difficulties are, and what kind of conditions we have to impose on the given functions, we first show some basic ideas.

1.8. The classical method. The following general method goes back to Andrade (1900) and Kac (1937) and is well known among functional equa- tionists. It is described (in a somewhat less general form) in the book of Aczdl [3], 4.2.2, 4.2.3. It can be applied to functional equations of the form

n

(1) h(x, ~ ) f (x) = ~ o ( x , Y) + c hi (x1 Y, fi ( ~ i ( x , Y ) ) ) ( 2 1 Y) t D; i=l

where f , f l , . . . , fn are the unknown functions defined on some open sets X, X I , . . . , Xn c R and taking values in some open sets Z, Z1,. . . , Zn c R. Let us suppose that Y C R is an open set, D C X x Y is also open, h, ho :

ah aho ahi ahi D 4 R and hi : D x Zi + R are continuous, and --, - - - fi ,

ax ax ' dx ' du ' i = 1,2, . . . , n are also continuous. Let us suppose that the functions ii are twice continuously differentiable. Let xo E X and suppose that there exists

agi a yo E Y such that (x0,yo) E D, ~ ( X O , yo) # 0, and -(xo,yo) # 0 for ay

i = 1,2, . . . , n. Then f is continuously differentiable in a neighborhood of 20.

To prove this, let us observe that for some open bounded interval Xo and for an interval [a, b] c Y we have that Xo x [a, b] C D, (xO, yo) E Xo x ]a, b[, and J~~ h(x, y) dy # 0 whenever x E Xo. Integrating both sides of equation (1) with respect to y over [a, b] we obtain


Using the notation gi,,(y) = gi(x, y) let us introduce the new variable u = gi,% (y) , i = 1 ,2 , . . . , n separately in each of the integrals on the right-hand side. For this we substitute Xo with a smaller neighborhood of xo and [a, b] with a smaller neighborhood of yo, if necessary. Then we obtain

From this equation it follows that f is continuously differentiable. Indeed, a parametric integral of the type

is continuously differentiable if 2, Y c R are bounded open sets, 3 : x x [a, b] + ? and ?2. : x x ? + R are continuous functions, and the partial

derivatives and 2 are continuous, too. To see this, let yl E ? be fixed, and let us denote

8 ( x , y) = l: L(x, u) du.

Then H is continuously differentiable and hence f (x) = H (x, g(x, b)) - H (x, g(x, a)) is also continuously differentiable.

ah Let us observe that we only need the continuity of and z. Therefore,

h(x, y) can have the form h* (x, y, fo(y)) with continuously differentiable h* but merely continuous (maybe unknown) fo which can even be vector valued. In particular, we may take h(x, y) = hi (x, y) f j (y) with m < 0 and with (possibly unknown) continuous functions f j , m 5 j < 0 and continuously differentiable functions hi. Moreover, let us observe that equations like

may be reduced to (1) introducing new variables x = G(Z, Q ) , y = Go(2, Q ) .


The generalization of this method to complex or even vector-valued unknown functions is immediate, but the generalization to vector-vector unknown functions is non-trivial. We can integrate over a simplex S with nonvoid interior, and we obtain the parametric integral

Of course, the differentiability of such parametric integrals is classic, if 3 and ?L are smooth, but we only know that 3 is continuously differentiable and h together with its partial derivative with respect to x is continuous. We investigate these questions and the generalization of the above classic method to vector-vector functions in 5 11.

1.9. Higher order differentiability. The above classical method can be used to prove higher order differentiability, if we use the fact that a parametric integral

I"(x) =

with smooth 3 is p + 1 times differentiable if h is p times continuously differentiable and its p th derivative is continuously partially differentiable with respect to x. An even better trick, which works also for the more general nonlinear functional equation 1.7.(1) is to take the p t h derivative of both sides of the equation. What we get is of type (1) from the previous point, and so can be treated by the above mentioned generalization of the classical method. This trick will be applied in 5 15.

1.10. Continuity of solutions. The classical method (and its generalization too) can also be applied to derive continuity of solutions from their local integrability. Hence, if we can obtain local boundedness of solutions from their measurability, then we can obtain continuity, too. This was the usual way to obtain continuity from measurability. For example, by Cauchy's equation f (x + y) = f (x) + f (y), if a solution f : R + R is measurable on a set K having positive Lebesgue measure, then it is automatically bounded on a smaller set C still having positive measure, because the measure of the sets K, = {x E K : I f (x)l 5 n) goes to the measure of K as n + m. Now by a classical theorem of Steinhaus, the set C - C contains a neighborhood of the origin. Function f is bounded on this neighborhood because if x E C - C, then x can be represented as x = z -y , z ,y E C, i. e., there exists a y E C such that x + y E C, too. But hence


i. e., f is bounded on a neighborhood of the origin. Now by integrating over [-E, E] we obtain

This implies that f is continuous near the origin, and by applying (1) again we get that f is differentiable near the origin. But by Cauchy's equation, fixing any y we see that f around y is merely a constant plus a translate of f around the origin. Hence if a solution is continuous, differentiable, analytic, etc. near the origin, then the same holds everywhere.

This way to get all measurable solutions f : R 4 R also works for Cauchy's exponential equation.

Today this method is much less important as was earlier, and we use it only if the measures in question are not Radon measures. Some application of this method for invariant expansions of the Haar measure and for covariant expansions of the Lebesgue measure will be given in $5 5, 6.

There is a much more appropriate method, which works for Radon measures, specially for the Lebesgue measure. It is based on using Lusin's theorem: an idea which seems to have first appeared in McKiernan [I591 and in Baker [29]. We have seen an example of this method in 1.3. The main idea works also for the general case

i. e., for equation 1.7.(2): Let us choose some compact set C with positive measure. Choosing a "large" compact subset Ki of gi (xo, C) on which fi is continuous, if we are able to prove that the set

is nonvoid whenever x is close enough to xo, then by substituting this y into the equation we obtain that f (x) is close to f ( x 0 ) We will use this method to obtain continuity directly from measurability: see § 8.

An important observation here is that it is enough if the functional equation is satisfied almost everywhere (see $ 10). The importance of this observation lies in that if the solutions are differentiable almost everywhere, then the derivative (which is automatically a Bore1 function, see 14.1) satisfies also a functional equation, and utilizing the above observation we obtain that the derivative is continuous. This idea, namely that "differentiability


almost everywhere implies continuous differentiability" is treated in 5 14. Let us observe that this means that from having locally bounded variation (for real variable solutions) or from having local Lipschitz continuity (for vector variable solutions) we can conclude continuous differentiability.

An interesting, but out-of-the-main-stream question is treated in 5 7. Problem 24 of Mazur in the Scottish Book [I571 asks whether additive func- t ional~ on a Banach space that are Lebesgue measurable along any curve are continuous. This is some kind of "measurability implies continuity" problem. We will briefly discuss it, proving that being Lebesgue measurable along any curve is equivalent to being universally measurable.

1.11. Steinhaus type theorems. As we have seen in the previous point, the theorem of Steinhaus [I891 is a useful tool to treat measurable solutions of functional equations. Another version asserts that, for any Lebesgue measurable sets Al, A2 C IR with positive Lebesgue measure, Al + A2 has an interior point. This theorem allows various generalizations and modifications; IR may be replaced by other topological measure spaces and the addition may be replaced by a two variable function. A large part of these generalizations are based on Weil's idea [206] that the convolution

is continuous. From this, by setting fi to the characteristic function of Ai the theorem of Steinhaus is easily obtained. Let us observe that the p measure of set 1.10.(1) in the previous point is

where fi is the characteristic function of Ki. This shows that what we need to prove "measurability implies continuity" type theorems is some kind of a generalized Steinhaus theorem, and the two topics are strongly connected.

In 5 3 we first generalize Weil's theorem proving the continuity of a

Using this result, we generalize the theorem of Steinhaus for the case of continuously differentiable functions of more than two variables. Several previously known results follow as special cases. This generalization will be used implicitly in $5 5, 8, 10, and 19 and explicitly in 5 6 and in the application given in 5 23.

2 2 Chapter I. Preliminaries

1.12. Baire category. There is a strong analogy between measure and Baire category: the book of Oxtoby [I611 is a nice introduction. Based on this, we may expect that most of the measure theoretical results have the category analogue. The category analogue of Steinhaus' theorem is the theorem of Piccard, stating that if C C R is of second category and has the Baire property, then C - C contains a neighborhood of the origin. This has similar generalizations as the theorem of Steinhaus: see $ 4. "Baire property implies continuity" type results are analogous to ''measurability implies continuity" type results; they are treated in $ 9.

1 . l3 . Differentiability of continuous solutions. No doubt, the hard- est step is to prove differentiability of solutions from continuity. As we have seen in 1.10, it is enough to derive differentiability almost everywhere; this is implied if we prove local Lipschitz property. The generalization of the classical method 1.8 for vector-vector functions is useful but far from being satisfying. We will investigate some properties between continuity and local Lipschitz property: Holder continuity (5 12) and essential bounded variation (5 13). Recently with some additional compactness conditions the author obtained a theorem, which is strong enough to use in most of the practical cases: see 11.6. The first application of this new result was the solution of the equation of "the duplication of the cube" 22.15. But I think the best illustration of the power of this result is that the recent results of the "Habil- itationsschrift" of M. Bonk about the characterization of Weierstrass's sigma function by functional equation (a problem which goes back to Abel) can be obtained, even in a more general setting; see $24.

1.14. Analyticity. The final step could be to prove that infinitely many times differentiable solutions are analytic. In this step there are only weak results; see $ 16. We will consider there a result from JBrai [95]. Often it is possible to derive &om the functional equation a differential equation. In such cases, regularity theory of differential equations can be used. See Dieudonni! [49], 10.5.3 for ordinary differential equations and the references in Zeidler [209], II/A, pp. 86-93 and the books [73], [74] of Hormander for partial differential equations. Such type of method is used in the paper of Lawruk and ~wia tak [I381 to prove analyticity of the solutions of a generalized mean value type equation. PBles [I631 also used differential equations to deduce analyticity on a "large" set; see the details in $ 16.

1.15. Other regularity properties. As we have seen above, steps (I)- (VI) from 1.7 give a natural "stairway" to climb up from measurability or Baire property to analyticity. Of course, several other regularity properties could be imagined. Let us briefly mention some of them.

Date post:	25-Jun-2020
Category:	Documents
Upload:	others
View:	0 times
Download:	0 times

REGULARITY PROPERTIES OF FUNCTIONAL EQUATIONS IN … · 2013-07-18 · PREFACE This book is about...

Documents